Definition 19.1.7 (Induced $\sigma$-Algebra).label Let $X$ be a set, $\cm \subset 2^{X}$ be a $\sigma$-algebra over $X$, and $E \subset X$, then the collection
\[\cm_{E} = \bracs{A \cap E|A \in \cm}\]
is the $\sigma$-algebra on $E$ induced by $\cm$.
Definition 19.1.7 (Induced $\sigma$-Algebra).label Let $X$ be a set, $\cm \subset 2^{X}$ be a $\sigma$-algebra over $X$, and $E \subset X$, then the collection
is the $\sigma$-algebra on $E$ induced by $\cm$.
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